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Batchelor vortex - Revision history
2026-04-07T08:00:55Z
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2013-03-14T21:17:06Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:17, 14 March 2013</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Irwin Butts </del>is <del style="font-weight: bold; text-decoration: none;">what my spouse enjoys </del>to <del style="font-weight: bold; text-decoration: none;">call me although I don</del>'<del style="font-weight: bold; text-decoration: none;">t truly like becoming known </del>as <del style="font-weight: bold; text-decoration: none;">like </del>that. <del style="font-weight: bold; text-decoration: none;">Years in </del>the <del style="font-weight: bold; text-decoration: none;">past we moved to North Dakota</del>. <del style="font-weight: bold; text-decoration: none;">Hiring </del>is <del style="font-weight: bold; text-decoration: none;">her day occupation now and she will not change </del>it <del style="font-weight: bold; text-decoration: none;">anytime quickly</del>. <del style="font-weight: bold; text-decoration: none;">To do aerobics </del>is a <del style="font-weight: bold; text-decoration: none;">factor that I</del>'<del style="font-weight: bold; text-decoration: none;">m totally addicted to</del>.<<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">Also visit my web page :: </del>[<del style="font-weight: bold; text-decoration: none;">http:</del>//<del style="font-weight: bold; text-decoration: none;">kpchurch</del>.<del style="font-weight: bold; text-decoration: none;">org/?document_srl</del>=<del style="font-weight: bold; text-decoration: none;">1359143 kpchurch</del>.<del style="font-weight: bold; text-decoration: none;">org</del>]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In [[mathematics]], the '''prime decomposition theorem for 3-manifolds''' states that every [[compact space|compact]], [[orientability|orientable]] [[3-manifold]] </ins>is <ins style="font-weight: bold; text-decoration: none;">the [[connected sum]] of a unique ([[up </ins>to<ins style="font-weight: bold; text-decoration: none;">]] [[homeomorphism]]) collection of [[prime manifold|prime 3-manifold]]s.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A manifold is </ins>'<ins style="font-weight: bold; text-decoration: none;">'prime'' if it cannot be presented </ins>as <ins style="font-weight: bold; text-decoration: none;">a connected sum of more than one manifold, none of which is the sphere of the same dimension. This condition is necessary since for any manifold M of dimension <math>n</math> it is true </ins>that</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>M=M\#S^n</ins>.<ins style="font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(where ''M#S<sup>n</sup>'' means </ins>the <ins style="font-weight: bold; text-decoration: none;">connected sum of ''M'' and ''S<sup>n</sup>'')</ins>. <ins style="font-weight: bold; text-decoration: none;">If ''P'' </ins>is <ins style="font-weight: bold; text-decoration: none;">a prime 3-manifold then either it is ''S''<sup>2</sup>&nbsp;&times;&nbsp;''S''<sup>1</sup> or the non-orientable ''S''<sup>2</sup> [[fiber bundle|bundle]] over ''S''<sup>1</sup>,</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">or </ins>it <ins style="font-weight: bold; text-decoration: none;">is [[irreducible manifold|irreducible]], which means that any embedded 2-sphere bounds a ball</ins>. <ins style="font-weight: bold; text-decoration: none;"> So the theorem can be restated to say that there </ins>is a <ins style="font-weight: bold; text-decoration: none;">unique connected sum decomposition into irreducible 3-manifolds and fiber bundles of ''S''<sup>2</sup> over ''S'</ins>'<ins style="font-weight: bold; text-decoration: none;"><sup>1</sup></ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The prime decomposition holds also for non-orientable 3-manifolds, but the uniqueness statement must be modified slightly: every compact, non-orientable 3-manifold is a connected sum of irreducible 3-manifolds and non-orientable ''S''</ins><<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">2</ins><<ins style="font-weight: bold; text-decoration: none;">/sup</ins>> [<ins style="font-weight: bold; text-decoration: none;">[fiber bundle|bundle]]s over ''S''<sup>1</sup>. This sum is unique as long as we specify that each summand is either irreducible or a non-orientable&nbsp;''S''<sup>2<</ins>/<ins style="font-weight: bold; text-decoration: none;">sup> [[fiber bundle|bundle]] over&nbsp;''S''<sup>1<</ins>/<ins style="font-weight: bold; text-decoration: none;">sup>.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The proof is based on [[normal surface]] techniques originated by [[Hellmuth Kneser]]. Existence was proven by Kneser, but the exact formulation and proof of the uniqueness was done more than 30 years later by [[John Milnor]]</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==References=</ins>=</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* J. Milnor, ''A unique decomposition theorem for 3-manifolds'', [[American Journal of Mathematics]] 84 (1962), 1&ndash;7</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:3-manifolds]</ins>]</div></td></tr>
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en>Addbot
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en>Lockley: remove context tag
2012-05-24T23:25:31Z
<p>remove context tag</p>
<p><b>New page</b></p><div>Irwin Butts is what my spouse enjoys to call me although I don't truly like becoming known as like that. Years in the past we moved to North Dakota. Hiring is her day occupation now and she will not change it anytime quickly. To do aerobics is a factor that I'm totally addicted to.<br><br>Also visit my web page :: [http://kpchurch.org/?document_srl=1359143 kpchurch.org]</div>
en>Lockley